Multi-objective Optimization

Many problems trade off several goals at once — accuracy against latency, return against risk. A multi-objective study optimizes a vector of objectives and returns the Pareto front: the set of trials where no objective can be improved without worsening another. OptunaSharp has no separate multi-objective module; it is the same Study API with more than one direction, mirroring Optuna.

Defining multiple objectives

Pass directions (plural) to CreateStudy and return an IReadOnlyList<double> from the objective — one value per direction, in order.

using OptunaSharp;

var study = Optuna.CreateStudy(directions: new[]
{
    StudyDirection.Minimize,   // objective 0: model size
    StudyDirection.Maximize,   // objective 1: accuracy
});

study.Optimize(trial =>
{
    long units = trial.SuggestInt("units", 16, 256);
    double drop = trial.SuggestFloat("dropout", 0.0, 0.5);

    double size = ModelSize(units);
    double acc  = Train(units, drop);
    return new[] { size, acc };     // IReadOnlyList<double>, aligned with directions
}, nTrials: 200);

The default sampler for a multi-objective study is NSGAIISampler.

Reading the Pareto front

Single-objective conveniences (BestValue, BestParams, BestTrial) don't apply — there is no single best. Use BestTrials, the non-dominated set:

foreach (FrozenTrial t in study.BestTrials)
    Console.WriteLine($"#{t.Number}: [{string.Join(", ", t.Values!)}]");

IReadOnlyList<StudyDirection> dirs = study.Directions;   // the objective directions

Each FrozenTrial.Values holds that trial's objective vector (versus Value for single-objective).

Choosing a sampler

NSGAIISampler (default) — non-dominated sorting with crowding distance; strong for 2–3 objectives.
NSGAIIISampler — reference-point based; better for many objectives.
TPESampler — drives MOTPE automatically when the study has multiple directions.
GPSampler — Gaussian-process multi-objective acquisition (log-EHVI and its constrained/batched variants) for expensive objectives.

See Samplers for construction and options.

Naming and constraints

Label the objectives for readable reports and visualizations:

study.SetMetricNames(new[] { "model_size", "accuracy" });

For constrained multi-objective optimization, supply a constraintsFunc to the sampler (supported by NSGAIISampler, NSGAIIISampler, TPESampler, and GPSampler); a trial is feasible when every returned constraint value is ≤ 0.

Visualizing trade-offs

Two visualization builders target multi-objective studies:

ParetoFront → ParetoFrontInfo — the front in objective space.
HypervolumeHistory → HypervolumeHistoryInfo — dominated hypervolume over time, relative to a reference point, a standard convergence measure.

Relationship to Optuna

Optuna (Python)	OptunaSharp
`create_study(directions=["minimize", "maximize"])`	`Optuna.CreateStudy(directions: new[] { StudyDirection.Minimize, StudyDirection.Maximize })`
`return [size, acc]`	`return new[] { size, acc }`
`study.best_trials`	`study.BestTrials`
`study.set_metric_names([...])`	`study.SetMetricNames(new[] { ... })`

Divergence. The hypervolume subset-selection (HSSP) can order zero-contribution ties differently for 3-or-more objectives — a documented, tested edge case in the shared engine.

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